Correct Answer: Option E

Explanation

To evaluate the expression \( \log 18 + \log 6 - \log 16^{\frac{1}{2}} \), we can use the properties of logarithms.

Simplify \( \log 16^{\frac{1}{2}} \)

Using the power rule of logarithms: \(\log 16^{\frac{1}{2}} = \frac{1}{2} \log 16\)

Substitute back into the expression:

Now rewrite the expression: \(\log 18 + \log 6 - \frac{1}{2} \log 16\)

Combine the logarithms:

Using the product property of logarithms: \(\log 18 + \log 6 = \log (18 \cdot 6) = \log 108\)

So we can rewrite the expression as: \(\log 108 - \frac{1}{2} \log 16\)

 Rewrite \( \frac{1}{2} \log 16\)

Using the power rule again: \(\frac{1}{2} \log 16 = \log 16^{\frac{1}{2}} = \log 4\)

Substitute back;

Now we have: \(\log 108 - \log 4\)

Using the quotient property of logarithms:  \(\log \left( \frac{108}{4} \right)\)

Simplify \(\frac{108}{4}\)

Calculating: \(\frac{108}{4}\) = 27

Thus, the expression simplifies to: log 27 = 3log3

NOTE: all is in base 10.

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